Shares of both companies surged on Friday, when Pershing said in regulatory filings it has a 9.77 percent stake in common shares of mortgage insurer Freddie Mac and a 9.98 percent stake in Fannie Mae.

Bruce Berkowitz of Fairholme Capital Management announced this week that he and other investors were willing to buy and recapitalize government-controlled Freddie Mac and its sister company, Fannie Mae.

In light of the proposed Fairholme transaction, Pershing said in the filings that "they may engage in discussions with management, the board, other stockholders of the issuer, representatives of the federal government, and other relevant parties" involved with Freddie Mac and Fannie Mae.

Pershing, which has more than $11.45 billion in assets under management, said the shares of Freddie Mac and Fannie Mae are "undervalued" and represent an "attractive investment."

The Federal Housing Finance Agency, which oversees Fannie and Freddie, declined to comment Friday on the Pershing Square investments. In a statement Friday, a Treasury official said, "The administration remains committed to reforming the housing-finance sector by responsibly winding down (Fannie and Freddie) and ensuring that any new system preserves broad access to credit for responsible borrowers, strengthens the economy and promotes financial stability."

Any effort to recapitalize Fannie Mae and Freddie Mac, which were seized by the U.S. government during the 2008 housing crisis, would require congressional approval. The White House and Congress have shown no interest so far in plans proposed by private investors.

In the summer of 2008, Ackman proposed a restructuring plan for Freddie Mac and Fannie Mae and said the mortgage companies should be moved to New York from Washington. At the same time, Ackman was betting against the shares of Freddie Mac and Fannie Mae. If the government had adopted Ackman's plan, it would have wiped out the common shares and preferred and would have given his short position a big return.

Fannie Mae and Freddie Mac are "something that I never thought should exist," said Margaret Patel, senior portfolio manager at Wells Capital Management. "It's awfully hard, having seen them both go under, to change that opinion," she added.

Patel said she would rather invest in the securities of banks with more proven track records than Fannie and Freddie.

Berkowitz's Fairholme Funds scooped up preferred shares of Fannie Mae and Freddie Mac with a face value of $3.5 billion at a massive discount, as well as some common shares.

Berkowitz's plan consists of contributing those shares as part of his proposed buyout of the insurance businesses of Fannie Mae and Freddie Mac. In the proposal obtained by Reuters, the businesses would be acquired in exchange for preferred shares at their full par value, or $34.6 billion. Berkowitz and investors including hedge fund-firms Paulson & Co. and Perry Capital LLC bought the preferred shares on bets of a recapitalization.

Fairholme's proposal would raise at least $17.3 billion in additional funds from preferred holders and through a rights offering.

A portfolio manager that specializes in mortgage credit, and who spoke on the condition of anonymity, questioned Ackman's decision to make such a large bet on Fannie and Freddie, calling the common shares and preferred shares "worthless." He said a plan to privatize the two companies "will never fly in Congress." He added that the government's credit guarantee has been the reason "for the profits as it was for the losses."

Fannie Mae and Freddie Mac have been operating under government conservatorship since the fall of 2008 and the 2012 changes of the bailout terms mean the two pay nearly all of their profits to the federal government in the form of dividends.

The two mortgage giants are close to paying back much of the $187 billion they received in a taxpayer-funded bailout and still play a dominant role in the U.S. mortgage market by backing the vast majority of newly issued home loans.